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Let me formally ask the question again.

Question. Is there a $T_{0}$-space that has a subset $A \subseteq X$ that has the folloing property?

There exists $p \in X$, a limit point of $A$ such that there is a neighborhood of $p$ intersects $A$ at finitely many points.

Note that there is no such subset $A$ in $T_{1}$ space.

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up vote 1 down vote accepted

Sure: the Sierpiński space is a simple example.

This is the set $\{0,1\}$ with the topology $\{\varnothing,\{1\},\{0,1\}\}$; $0$ is a limit point of $\{1\}$.

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Oh gee. I came up with the same example but somehow thought that I had to check for all the neighborhoods. Thanks, now I even know the name of the example! – user123454321 Oct 19 '12 at 4:48
You’re welcome! – Brian M. Scott Oct 19 '12 at 4:49

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